2
Task 1
:
Derive the mathematical model of the system
Denote the mass of the cart as
m
, the mass of the pendulum as
M
, the length of the
pendulum
L
, and the stiffness of the spring as
k
.
The
values of
parameters are
m
= 0
.
455
kg,
M
= 0
.21 kg,
L
= 0
.61
m
,
k
= 100 N/m,
k
1
=
1.7
2
N/V
, and
k
2
=
7.68
Ns/m
.
Assume
that the disturbance force
w
is applied
at a distance of 2
L
/3 from the cart
–
pendulum hinge.
a.
Draw the necessary free
–
body diagram
and derive the nonlinear equations of
motion.
b.
Linearize the equations of m
otion for small angular motions.
c.
D
etermine the state
–
space form with
p
and
θ
as the outputs.
Task 2: Build a Simulink model for the system
The system
is
disturbed by a sharp tap on
the pendulum that comes from a human hand.
For simulation purposes, the disturbance force
w
is m
odeled as a const
ant force of 1
6
.
0
N
with duration of 0.01 sec,
a
nd
a
ssume that
u
=0
.
a.
cart
–
spring
–
p
endulum system is described by the state
–
space model derived in
Task 1(b).
b.
Build a Simulink block diagram for nonlinear simulation, where the dynamics of
the cart
–
spring
–
pendulum system is described by the nonlinear differential
equations derived in Task
1(a).
Task 3: Analyze the response of the system
Run the above two Simulink files, plot the
following
time responses
and compare the
a.
The position of the cart
p
vs. time
b.
The angular position of the cart
vs. time
Groups
: This is a group design project. Each student may choose his/her own group, and
the group size is not greater than
3
.
Grading:
Grades will be based on a combination of group performance and individual
contributions. Group performance will be based on the quality of work contained in the
report and the involvement of all team members.
Report
: The required contents include
a.
Introd
uction
b.
Mathematical modeling of the system
c.
Construction of Simulink block diagrams
d.
Linear and nonlinear simulation results
e.
Discussions